UC Irvine

A pervasive and often tacit assumption in generative modeling is that our data distribution is finite-dimensional. These finite-dimensional distributions frequently arise from a discrete representation of some continuous underlying signal. Images, for instance, are represented as a finite collection of pixels, and generative models are typically built directly on top of this pixel-level representation. However, our world is not made of pixels, and building faithful models of our world requires moving beyond this assumption. This is particularly true for data modalities like partial differential equations and time series, where the multi-scale or irregularly sampled nature of this data is a key feature. In this dissertation, we develop both the theory and methodology necessary to build generative models for infinite-dimensional data. In particular, we focus on a class of models which can be understood through the lens of measure transport. %These models, including flows and diffusions, can be seen as parametrizing a curve in the space of measures which approximately interpolates between a fixed reference measure and our data measure.

We begin with an overview of this class of models and a review of the necessary mathematical background. We build upon this background to develop techniques for building diffusion and flow-based generative models for infinite-dimensional data. Our focus then shifts to conditional generation tasks and the application of optimal transport techniques within flow-based models. Finally, we apply flow-based techniques to forecast continuous-time event data before concluding with a discussion of several remaining challenges.